Causal Effects and Experiments

GVPT399F: Power, Politics, and Data

Causes and effects

Our goal is to better understand what factors lead to certain outcomes of interest.

  • Does increasing the number of voting booths close to a potential voter make that person more likely to vote?

  • Do peace treaties signed with factionalized rebel groups more often lead to a return to conflict than those signed with a single, cohesive group?

  • Does trade between two countries make war between them less likely?

Causes and effects

These are causal statements:

  • More local voting booths \(\rightarrow\) More likely to vote

  • More factionalization \(\rightarrow\) More likely to restart conflict

  • Trade \(\rightarrow\) Less likely to go to war

How to identify causes and their effects

Proving that changes to one factor cause changes to another is very tricky!

  • Need to account for all the other factors

The efficacy of international election monitors

Do international monitors cause less election-day fraud in democratic elections?

  • Number of elections monitored by international observers exploded throughout the 2000s

  • But do monitors increase the chances the election will be free and fair?

  • In other words, are they effective?

The efficacy of international election monitors

Dr Susan Hyde set out to answer this very question. From her article (page 39):

If the presence of international observers causes a reduction in election-day fraud, the effect of observers should be visible at the subnational level by comparing polling stations that were visited by observers with those that were not visited. More specifically, if international monitoring reduces electionday fraud directly, all else held equal, the cheating parties should gain less of their ill-gotten vote share in polling stations that were visited by international monitors.

Causal relationships

Refers to the directional connection between a change in one variable and a corresponding change in another.

  • The direction matters!

  • Treatment variable: the variable causing changes to another variable.

  • Outcome variable: the variable changing as a result of changes to another variable (the treatment).

Treatment and outcome variables

We want to test whether the presence of international monitors (treatment) leads to less election day fraud (outcome).

International election monitors present at polling stations \(\rightarrow\) Less election-day fraud at that station

Treatment variable

At any given polling station in any given election, monitors may be: 1) present, or 2) not present.

Two different conditions:

  • Treatment: condition with treatment (monitors are present)

  • Control: condition without treatment (monitors are not present)

Outcome variable

At any given polling station in any given election, fraud may: 1) occur, or 2) not occur.

Observing our outcome

Sometimes it can be hard to observe our outcome of interest.

  • How might we see all election-day fraud?

  • How might we measure it?

Observing our outcome

Hyde’s answer: vote-share!

International election monitors present at polling stations \(\rightarrow\) Less election-day fraud at that station \(\rightarrow\) Lower vote-share for the cheating party

Individual causal effects

We want to know whether the treatment causes a change in our outcome of interest.

What might this look like in the real world?

  • Imagine we are looking at a specific election

  • Five polling stations

Parallel worlds

Hypothetical election

ID Monitored vote % Non-monitored vote %
1 79.96 93.13
2 49.23 90.33
3 37.81 87.30
4 29.19 76.77
5 33.97 84.32

Individual effects

The only difference between these two conditions is the presence of international monitors!

  • The difference between vote shares under these conditions is caused by the monitors.

Individual effects

ID Monitored vote % Non-monitored vote % Difference (%)
1 79.96 93.13 -13.17
2 49.23 90.33 -41.10
3 37.81 87.30 -49.49
4 29.19 76.77 -47.58
5 33.97 84.32 -50.35

No parallel worlds

Sadly for us, we cannot create parallel worlds…

What now?

ID Monitored Monitored vote % Non-monitored vote % Difference
1 0 NA 93.13 NA
2 1 49.23 NA NA
3 0 NA 87.30 NA
4 1 29.19 NA NA
5 0 NA 84.32 NA

Average causal effects

We need to move away from looking at individuals and start to look for patterns in our group.

Back to our parallel worlds

ID Monitored vote % Non-monitored vote % Difference
1 79.96 93.13 -13.17
2 49.23 90.33 -41.10
3 37.81 87.30 -49.49
4 29.19 76.77 -47.58
5 33.97 84.32 -50.35

Difference of averages across all individuals

What was the average vote share received in each world?

Avg. monitored vote % Avg. non-monitored vote % Difference
46.03 86.37 -40.34

Back to reality

ID Monitored Monitored vote % Non-monitored vote % Difference
1 0 NA 93.13 NA
2 1 49.23 NA NA
3 0 NA 87.30 NA
4 1 29.19 NA NA
5 0 NA 84.32 NA

Difference-of-means

What was the average vote share received in each group?

Avg. monitored vote % Avg. non-monitored vote % Difference
39.21 88.25 -49.04

How on Earth does this work so well?

Randomization!

  • Monitors were assigned to polling stations randomly (for example, with the flip of a coin)

  • This created two groups of stations that were roughly identical on average to one another prior to treatment

  • This mimics what happens when we split our world into two (creating two literally identical groups)

Our goal

We want two groups that are completely identical to one another prior to treatment.

  • This allows us to compare their outcomes after treatment and claim that the treatment caused differences in those outcomes (if any differences exist)

Our goal

Next best thing

Instead, we should try to make two groups that are as similar as possible to each other prior to treatment.

The magic of randomization

Perfectly random assignment does this very well!

Don’t take my word for it

Imagine we have a group of 1,000 individuals. We know the following about them:

  • Height

  • Weight

  • Eye colour

Our group

ID Height Weight Eye colour
1 169.67 66.25 Grey
2 173.07 77.49 Blue
3 177.22 84.55 Blue
4 161.48 68.89 Blue
5 182.28 79.94 Blue
6 166.34 77.79 Grey
7 174.67 77.06 Brown
8 177.08 84.17 Brown
9 165.39 67.19 Brown
10 170.43 86.49 Blue
11 173.90 81.97 Brown
12 163.88 91.35 Blue
13 170.08 90.36 Brown
14 176.58 69.89 Grey
15 163.34 84.29 Grey
16 171.20 64.83 Green
17 168.31 72.28 Blue
18 173.78 85.94 Grey
19 174.38 93.37 Grey
20 167.46 60.09 Green
21 179.75 80.55 Grey
22 168.72 55.62 Grey
23 163.97 69.88 Green
24 167.42 64.65 Blue
25 175.88 65.71 Grey
26 169.16 70.00 Blue
27 178.06 81.75 Green
28 165.62 61.84 Green
29 162.57 86.71 Green
30 157.92 82.34 Green
31 160.98 82.65 Green
32 169.79 92.62 Grey
33 163.95 93.35 Blue
34 170.50 80.65 Green
35 169.28 98.70 Green
36 174.00 80.39 Green
37 169.37 105.51 Brown
38 163.08 79.24 Blue
39 164.63 87.55 Blue
40 173.05 70.55 Brown
41 176.38 81.30 Blue
42 173.28 76.12 Brown
43 153.34 80.19 Brown
44 176.71 88.35 Green
45 165.33 64.82 Grey
46 163.15 64.17 Grey
47 166.44 80.59 Blue
48 174.72 84.70 Grey
49 163.70 76.83 Blue
50 175.31 81.44 Brown
51 156.73 89.48 Green
52 167.73 80.08 Blue
53 179.72 87.96 Blue
54 169.92 85.19 Blue
55 171.18 99.64 Brown
56 155.76 88.90 Green
57 162.26 56.96 Grey
58 158.84 75.51 Grey
59 150.74 70.19 Green
60 176.27 70.51 Green
61 169.75 75.31 Blue
62 175.34 89.45 Grey
63 169.99 97.23 Green
64 164.14 87.03 Green
65 159.63 75.01 Blue
66 161.95 58.41 Green
67 164.31 93.66 Grey
68 153.77 80.86 Brown
69 171.99 89.15 Grey
70 167.41 89.92 Brown
71 165.72 90.26 Blue
72 166.25 78.84 Grey
73 183.10 66.15 Blue
74 167.66 79.84 Grey
75 171.15 86.23 Blue
76 167.40 71.65 Grey
77 169.46 90.00 Grey
78 155.28 77.01 Blue
79 163.38 59.35 Grey
80 178.22 72.58 Green
81 161.95 77.90 Grey
82 177.46 73.13 Grey
83 170.53 69.19 Green
84 168.86 89.00 Green
85 179.85 85.48 Grey
86 171.53 77.63 Green
87 175.24 75.91 Blue
88 181.81 94.81 Blue
89 164.16 79.06 Brown
90 160.76 89.50 Brown
91 161.75 89.30 Blue
92 168.40 80.54 Green
93 164.69 90.93 Blue
94 164.61 81.30 Blue
95 168.25 69.42 Grey
96 172.41 68.56 Blue
97 170.50 82.31 Blue
98 177.90 66.75 Green
99 170.18 91.41 Green
100 176.35 88.30 Grey
101 163.61 79.84 Grey
102 170.37 83.49 Blue
103 162.80 83.93 Grey
104 165.08 100.79 Green
105 168.06 77.16 Grey
106 178.30 87.79 Blue
107 158.56 74.17 Brown
108 176.90 90.69 Grey
109 165.05 83.44 Brown
110 176.26 88.39 Brown
111 164.35 94.24 Brown
112 162.52 89.67 Brown
113 163.37 82.60 Brown
114 168.13 85.49 Brown
115 167.09 82.06 Blue
116 166.98 75.99 Green
117 170.33 80.69 Brown
118 168.70 79.16 Grey
119 171.32 66.15 Blue
120 178.29 79.79 Blue
121 176.63 93.71 Grey
122 170.27 85.53 Grey
123 168.90 68.24 Brown
124 174.65 54.62 Green
125 173.43 71.15 Brown
126 163.25 84.85 Blue
127 173.17 74.42 Blue
128 159.67 69.37 Brown
129 170.33 88.54 Brown
130 176.08 70.47 Blue
131 165.73 87.17 Brown
132 175.71 64.11 Brown
133 158.95 90.90 Brown
134 172.04 94.32 Brown
135 176.93 82.80 Grey
136 166.87 94.96 Brown
137 168.25 83.90 Green
138 166.54 76.47 Grey
139 178.04 75.91 Green
140 171.63 85.71 Brown
141 174.82 87.38 Brown
142 170.00 69.44 Blue
143 165.06 91.37 Blue
144 163.98 91.13 Brown
145 170.54 62.06 Blue
146 164.56 63.25 Grey
147 165.24 83.45 Brown
148 169.56 76.95 Green
149 163.94 74.58 Brown
150 174.57 70.56 Grey
151 172.37 65.14 Brown
152 168.59 75.45 Grey
153 178.13 69.47 Brown
154 158.96 80.51 Grey
155 175.68 93.20 Grey
156 165.04 99.50 Brown
157 168.45 46.30 Brown
158 167.92 77.50 Grey
159 161.46 95.20 Brown
160 168.03 66.89 Blue
161 160.48 77.25 Brown
162 179.46 78.11 Brown
163 166.45 68.23 Brown
164 169.67 77.34 Green
165 177.31 89.58 Green
166 171.62 78.98 Brown
167 170.61 65.36 Green
168 166.52 69.72 Blue
169 167.49 62.28 Green
170 167.58 82.87 Blue
171 172.18 103.63 Grey
172 152.55 64.18 Brown
173 175.02 91.32 Blue
174 171.82 64.80 Blue
175 163.88 73.52 Green
176 169.69 80.78 Grey
177 166.43 75.20 Brown
178 174.37 72.38 Blue
179 173.27 72.09 Green
180 162.33 71.77 Green
181 167.34 72.77 Green
182 172.31 94.33 Grey
183 157.00 78.63 Brown
184 173.55 62.94 Brown
185 171.23 80.66 Grey
186 167.64 86.47 Blue
187 171.14 83.63 Grey
188 171.20 81.59 Green
189 171.83 86.94 Blue
190 174.15 74.26 Green
191 170.33 81.00 Grey
192 177.74 89.21 Brown
193 171.34 96.63 Blue
194 158.77 90.77 Green
195 173.11 77.73 Green
196 173.39 83.76 Grey
197 173.02 59.01 Blue
198 163.17 78.44 Grey
199 173.85 91.39 Blue
200 168.48 101.98 Brown
201 176.13 103.48 Green
202 162.57 91.50 Green
203 173.96 76.52 Grey
204 165.54 61.81 Grey
205 166.79 92.50 Brown
206 167.15 90.50 Green
207 170.88 72.43 Brown
208 176.89 70.32 Grey
209 173.79 83.16 Brown
210 161.67 93.15 Blue
211 180.41 62.05 Blue
212 168.30 75.67 Grey
213 169.95 71.30 Green
214 170.10 79.64 Green
215 169.62 74.89 Blue
216 165.53 72.20 Blue
217 174.74 76.32 Blue
218 165.04 58.08 Brown
219 169.19 93.89 Brown
220 171.67 83.03 Grey
221 180.26 93.00 Brown
222 173.77 88.11 Brown
223 171.43 79.40 Brown
224 165.79 79.97 Green
225 175.11 83.71 Green
226 159.92 91.34 Green
227 173.42 94.30 Blue
228 174.38 91.77 Blue
229 167.75 83.27 Blue
230 173.25 65.85 Blue
231 166.84 74.42 Grey
232 166.09 60.93 Green
233 170.41 91.47 Green
234 177.48 82.41 Grey
235 168.34 78.32 Grey
236 179.02 70.87 Brown
237 172.50 58.46 Green
238 164.71 79.70 Green
239 167.53 78.63 Blue
240 165.53 76.99 Brown
241 160.51 76.17 Blue
242 170.72 71.19 Blue
243 170.25 96.24 Grey
244 168.02 76.44 Brown
245 160.35 66.39 Blue
246 168.02 75.83 Brown
247 174.09 85.73 Grey
248 180.22 86.70 Green
249 178.29 86.30 Blue
250 168.96 86.36 Blue
251 171.80 87.82 Green
252 166.88 83.18 Blue
253 163.95 67.66 Green
254 165.55 51.50 Grey
255 165.44 73.29 Grey
256 168.18 82.85 Grey
257 173.37 104.05 Grey
258 171.92 91.95 Blue
259 177.11 65.89 Grey
260 164.24 85.79 Brown
261 162.73 77.69 Grey
262 171.02 66.53 Green
263 171.90 69.30 Green
264 178.69 82.23 Blue
265 171.34 89.01 Blue
266 162.81 74.98 Brown
267 164.44 92.84 Blue
268 176.29 75.55 Grey
269 179.40 77.74 Blue
270 167.72 76.49 Grey
271 169.35 79.56 Green
272 167.85 74.17 Blue
273 171.22 66.68 Grey
274 174.01 73.62 Blue
275 165.79 92.47 Green
276 170.44 84.38 Brown
277 171.79 75.18 Blue
278 167.62 76.82 Green
279 170.43 97.69 Grey
280 169.49 94.31 Brown
281 168.82 80.38 Green
282 163.19 74.14 Brown
283 172.74 64.72 Grey
284 172.99 85.37 Blue
285 181.01 64.75 Green
286 167.32 70.22 Green
287 161.76 79.12 Green
288 175.39 65.83 Blue
289 172.49 58.91 Grey
290 169.48 77.32 Brown
291 171.04 63.96 Blue
292 166.91 69.87 Green
293 172.16 72.75 Green
294 163.50 74.85 Brown
295 169.25 71.03 Blue
296 156.55 85.45 Green
297 176.93 89.49 Brown
298 172.39 74.21 Blue
299 175.32 80.00 Grey
300 173.70 70.07 Grey
301 162.02 94.04 Grey
302 165.75 65.25 Blue
303 157.00 85.62 Blue
304 176.94 77.30 Blue
305 183.07 80.75 Grey
306 164.46 63.66 Brown
307 170.59 74.69 Brown
308 174.93 83.28 Grey
309 167.43 58.90 Blue
310 178.23 77.13 Blue
311 173.08 82.83 Brown
312 177.13 72.76 Grey
313 165.98 72.69 Blue
314 159.16 82.02 Green
315 180.21 80.08 Grey
316 179.47 84.34 Grey
317 169.05 87.30 Grey
318 167.53 69.95 Grey
319 165.03 85.89 Green
320 178.33 84.27 Brown
321 163.15 57.84 Blue
322 174.94 69.65 Green
323 169.97 80.56 Brown
324 169.57 91.61 Brown
325 163.56 78.59 Brown
326 166.60 72.65 Green
327 167.92 79.59 Brown
328 165.29 77.95 Green
329 163.50 82.21 Brown
330 167.69 80.23 Brown
331 167.34 81.09 Brown
332 183.18 73.28 Grey
333 168.05 63.28 Brown
334 168.46 83.19 Brown
335 169.69 76.85 Brown
336 162.30 74.61 Brown
337 161.20 62.75 Brown
338 175.36 71.88 Green
339 175.72 60.90 Blue
340 169.14 85.45 Brown
341 171.49 76.97 Grey
342 165.65 85.93 Blue
343 170.67 76.84 Grey
344 161.02 90.57 Brown
345 172.42 95.07 Green
346 167.73 76.67 Grey
347 166.65 84.11 Green
348 185.80 74.56 Green
349 176.11 84.55 Brown
350 182.05 65.14 Blue
351 158.28 94.00 Blue
352 168.25 86.08 Blue
353 175.35 84.48 Brown
354 167.74 90.33 Blue
355 176.02 78.57 Blue
356 174.01 65.81 Blue
357 170.95 72.78 Brown
358 174.90 74.07 Blue
359 174.78 69.25 Green
360 180.85 78.87 Blue
361 171.75 74.72 Blue
362 170.16 63.45 Green
363 163.83 74.78 Brown
364 163.54 66.85 Green
365 166.24 63.27 Blue
366 168.59 75.87 Blue
367 175.89 88.61 Grey
368 162.23 83.82 Brown
369 163.71 89.80 Blue
370 169.93 78.73 Grey
371 177.18 90.98 Grey
372 170.24 75.03 Blue
373 174.87 75.69 Green
374 162.40 79.13 Green
375 170.13 79.79 Grey
376 166.94 74.00 Green
377 168.56 80.46 Green
378 174.01 88.67 Blue
379 170.75 69.65 Blue
380 164.57 68.81 Grey
381 166.38 83.07 Brown
382 173.81 103.64 Grey
383 176.74 83.21 Blue
384 172.77 86.78 Grey
385 167.69 83.88 Brown
386 169.92 64.74 Green
387 164.29 76.11 Brown
388 169.80 69.92 Blue
389 157.95 68.20 Blue
390 170.37 77.17 Brown
391 166.87 83.42 Blue
392 176.80 73.38 Blue
393 174.20 86.58 Brown
394 171.09 79.50 Grey
395 180.39 90.74 Blue
396 162.37 94.78 Brown
397 167.97 82.86 Grey
398 173.60 75.71 Brown
399 163.64 76.49 Grey
400 168.62 56.65 Brown
401 161.10 68.50 Brown
402 174.01 91.69 Grey
403 172.16 79.52 Green
404 177.43 81.52 Green
405 177.17 78.78 Blue
406 168.58 82.51 Grey
407 162.25 84.93 Blue
408 167.03 79.73 Green
409 171.97 75.60 Brown
410 165.83 70.53 Grey
411 155.46 80.38 Blue
412 161.78 83.76 Green
413 175.19 70.08 Blue
414 163.85 79.53 Brown
415 177.12 92.35 Brown
416 175.46 79.74 Grey
417 167.62 92.91 Green
418 176.03 92.59 Brown
419 163.96 80.21 Blue
420 168.65 89.09 Brown
421 172.76 78.41 Brown
422 171.72 93.61 Brown
423 170.68 76.29 Brown
424 176.70 82.33 Green
425 161.56 82.09 Brown
426 175.66 84.89 Grey
427 161.58 106.96 Grey
428 155.74 85.07 Green
429 168.39 80.94 Green
430 162.14 88.56 Blue
431 175.06 65.90 Grey
432 174.33 84.15 Blue
433 171.29 75.52 Blue
434 166.95 93.32 Blue
435 164.80 71.76 Green
436 166.99 85.78 Grey
437 176.20 92.28 Green
438 174.71 71.88 Brown
439 168.13 68.53 Grey
440 169.54 83.67 Brown
441 169.34 91.25 Grey
442 182.19 68.10 Brown
443 168.11 61.86 Grey
444 174.34 79.88 Blue
445 171.55 90.24 Green
446 169.41 72.22 Brown
447 172.31 86.63 Grey
448 160.99 71.93 Green
449 168.53 58.73 Brown
450 168.18 73.21 Grey
451 174.76 91.14 Grey
452 178.12 83.63 Blue
453 168.93 70.12 Green
454 170.28 92.61 Green
455 169.80 78.81 Brown
456 174.86 94.98 Grey
457 175.70 83.20 Blue
458 172.81 79.91 Blue
459 163.41 67.28 Green
460 158.94 67.60 Blue
461 170.82 93.56 Blue
462 164.87 88.40 Green
463 163.52 73.19 Blue
464 171.90 74.78 Green
465 166.24 93.86 Brown
466 171.65 83.36 Brown
467 175.13 74.10 Grey
468 166.62 70.00 Green
469 170.90 91.01 Green
470 176.05 78.23 Blue
471 171.29 77.75 Blue
472 172.61 73.95 Blue
473 178.86 72.93 Grey
474 172.50 84.50 Blue
475 163.77 85.30 Grey
476 166.20 79.78 Green
477 172.29 70.87 Brown
478 175.71 80.55 Blue
479 166.11 73.41 Green
480 168.12 87.81 Blue
481 167.60 79.10 Green
482 166.16 80.94 Blue
483 173.79 86.86 Brown
484 174.91 71.13 Brown
485 161.76 83.33 Brown
486 169.21 71.44 Brown
487 176.63 90.14 Blue
488 167.60 81.52 Brown
489 167.94 82.32 Blue
490 175.36 97.69 Blue
491 170.48 97.11 Green
492 159.35 77.57 Blue
493 172.98 87.22 Blue
494 167.08 88.95 Grey
495 183.28 84.62 Brown
496 171.71 72.31 Brown
497 175.19 82.13 Grey
498 165.91 74.58 Grey
499 172.28 77.61 Green
500 158.19 82.44 Blue
501 171.93 77.86 Grey
502 165.49 67.09 Brown
503 172.39 88.61 Grey
504 156.17 66.08 Green
505 165.22 77.81 Green
506 168.80 60.26 Grey
507 163.78 68.12 Grey
508 169.06 67.72 Blue
509 161.53 61.52 Brown
510 167.96 77.25 Blue
511 163.69 87.84 Grey
512 167.75 66.31 Blue
513 176.16 86.21 Brown
514 177.91 89.78 Brown
515 156.78 78.65 Grey
516 167.26 64.35 Brown
517 179.76 76.17 Grey
518 165.62 78.30 Blue
519 162.94 70.85 Blue
520 166.69 74.72 Green
521 173.34 89.88 Blue
522 168.59 89.67 Blue
523 176.38 88.78 Brown
524 166.84 79.39 Blue
525 163.21 94.70 Blue
526 165.76 81.70 Grey
527 170.15 95.34 Green
528 171.57 80.29 Brown
529 164.89 82.41 Blue
530 169.42 84.39 Brown
531 178.41 79.31 Grey
532 165.07 75.72 Green
533 162.92 83.84 Blue
534 174.29 83.60 Brown
535 163.15 76.57 Grey
536 165.51 79.69 Blue
537 170.40 86.27 Blue
538 170.53 73.89 Green
539 169.18 69.30 Brown
540 164.75 78.19 Grey
541 173.57 71.87 Grey
542 170.77 69.52 Grey
543 160.79 90.26 Blue
544 172.72 56.69 Green
545 163.43 100.84 Grey
546 171.23 88.51 Grey
547 174.08 80.74 Green
548 166.88 60.29 Green
549 178.43 77.01 Blue
550 166.30 73.07 Grey
551 167.15 77.73 Green
552 181.67 81.12 Brown
553 177.56 89.42 Green
554 162.68 90.49 Grey
555 177.65 74.99 Grey
556 167.79 63.24 Blue
557 175.02 89.23 Green
558 173.32 69.26 Green
559 164.91 71.33 Brown
560 174.35 91.37 Green
561 169.56 60.45 Grey
562 163.08 51.92 Green
563 167.36 86.34 Blue
564 176.27 65.28 Grey
565 160.89 67.87 Brown
566 181.94 72.99 Blue
567 167.86 79.87 Grey
568 171.58 76.01 Brown
569 182.72 78.96 Blue
570 166.58 82.91 Brown
571 172.64 68.26 Brown
572 168.52 76.11 Brown
573 177.49 87.15 Grey
574 179.59 63.62 Grey
575 176.72 69.82 Grey
576 162.78 60.89 Blue
577 177.53 78.34 Brown
578 170.44 81.64 Green
579 168.92 59.25 Green
580 169.79 85.94 Green
581 171.44 69.18 Grey
582 164.51 78.68 Green
583 168.27 76.40 Green
584 168.93 91.60 Green
585 166.47 82.87 Grey
586 160.76 65.36 Blue
587 169.52 75.73 Green
588 176.90 89.42 Grey
589 174.21 94.73 Green
590 187.87 68.18 Green
591 166.11 76.04 Green
592 176.90 48.41 Blue
593 170.32 79.63 Grey
594 160.28 73.36 Blue
595 180.06 83.45 Green
596 186.73 83.70 Green
597 167.46 86.99 Grey
598 170.29 89.85 Brown
599 160.97 76.77 Grey
600 173.71 74.82 Brown
601 167.63 86.52 Grey
602 169.09 66.55 Blue
603 159.16 88.91 Blue
604 168.37 91.49 Green
605 176.00 70.23 Blue
606 156.01 69.02 Brown
607 166.90 66.61 Grey
608 165.36 74.44 Brown
609 166.17 62.86 Green
610 176.70 87.29 Blue
611 177.42 78.15 Brown
612 164.72 99.82 Grey
613 162.33 70.30 Green
614 167.08 75.96 Blue
615 172.92 89.45 Blue
616 176.38 90.26 Brown
617 164.35 99.98 Green
618 161.33 89.42 Green
619 172.89 72.92 Blue
620 170.09 67.26 Blue
621 162.07 85.68 Blue
622 166.42 79.14 Green
623 164.99 86.53 Blue
624 174.66 97.71 Brown
625 171.69 83.08 Blue
626 181.11 82.09 Grey
627 174.95 90.11 Blue
628 158.99 77.31 Brown
629 158.09 82.81 Brown
630 174.77 97.32 Grey
631 169.11 93.13 Brown
632 172.59 93.14 Brown
633 169.70 79.31 Blue
634 167.12 82.27 Green
635 167.64 92.75 Green
636 174.27 72.74 Grey
637 172.07 75.33 Blue
638 164.73 79.23 Brown
639 173.21 66.91 Brown
640 171.36 90.01 Green
641 181.71 76.30 Grey
642 182.42 93.69 Blue
643 169.61 77.74 Green
644 169.94 80.28 Green
645 161.31 85.89 Brown
646 170.82 69.60 Grey
647 167.34 72.56 Grey
648 173.51 77.10 Grey
649 177.17 82.81 Grey
650 173.15 92.59 Green
651 181.22 85.25 Blue
652 162.52 74.60 Green
653 168.46 83.11 Brown
654 174.55 88.56 Brown
655 169.81 60.38 Grey
656 168.51 86.75 Brown
657 166.28 73.65 Brown
658 165.82 74.41 Grey
659 169.34 88.84 Blue
660 168.72 69.02 Brown
661 171.23 71.29 Green
662 153.97 84.77 Grey
663 166.65 88.47 Green
664 172.50 91.36 Blue
665 169.29 81.93 Grey
666 168.82 75.03 Grey
667 165.97 76.03 Brown
668 185.29 79.42 Brown
669 172.73 79.34 Brown
670 168.55 95.58 Blue
671 166.32 80.88 Grey
672 169.42 94.68 Grey
673 175.61 90.74 Blue
674 162.14 81.25 Green
675 168.29 85.46 Grey
676 174.46 81.54 Brown
677 172.01 66.11 Grey
678 173.39 82.46 Brown
679 163.49 86.31 Brown
680 173.13 86.92 Brown
681 175.27 78.20 Green
682 172.68 68.31 Grey
683 172.69 84.18 Blue
684 174.48 70.18 Brown
685 171.44 87.02 Brown
686 180.49 66.54 Blue
687 170.42 88.85 Blue
688 174.09 79.16 Grey
689 170.19 82.46 Brown
690 173.29 77.31 Green
691 177.17 72.70 Green
692 184.40 88.05 Grey
693 164.17 82.73 Green
694 176.87 50.04 Grey
695 163.74 77.81 Green
696 167.06 85.49 Green
697 166.98 92.84 Grey
698 178.55 73.59 Blue
699 167.90 100.46 Grey
700 172.32 75.31 Green
701 175.94 85.47 Green
702 174.88 73.51 Green
703 165.82 70.07 Brown
704 166.91 86.90 Brown
705 165.89 75.70 Blue
706 178.13 74.46 Grey
707 169.14 84.96 Green
708 171.26 93.53 Grey
709 179.73 88.67 Grey
710 174.96 69.16 Green
711 171.94 66.08 Grey
712 171.25 76.27 Brown
713 168.01 80.72 Green
714 170.76 102.41 Grey
715 172.41 98.88 Brown
716 164.50 81.08 Green
717 184.19 85.59 Grey
718 174.84 96.25 Grey
719 175.76 84.99 Blue
720 174.04 83.28 Blue
721 169.56 63.95 Grey
722 174.79 64.56 Brown
723 170.61 90.52 Green
724 167.94 75.20 Blue
725 166.65 83.80 Blue
726 166.59 87.99 Brown
727 162.11 70.68 Brown
728 162.95 71.83 Blue
729 174.61 70.39 Brown
730 169.53 80.73 Green
731 164.74 74.89 Brown
732 167.52 65.03 Grey
733 179.17 79.83 Brown
734 164.45 84.92 Blue
735 176.57 82.35 Grey
736 179.69 53.65 Brown
737 178.80 70.41 Green
738 169.55 95.39 Blue
739 171.04 90.30 Grey
740 159.18 84.80 Blue
741 167.39 97.14 Blue
742 171.59 94.06 Green
743 165.80 70.18 Blue
744 180.39 83.52 Blue
745 172.28 84.92 Grey
746 162.85 84.10 Grey
747 166.76 67.13 Green
748 172.28 70.42 Grey
749 167.07 74.62 Brown
750 177.60 76.73 Blue
751 166.38 81.71 Green
752 173.11 86.63 Green
753 168.69 72.60 Grey
754 182.63 92.33 Green
755 170.80 69.58 Grey
756 162.67 94.45 Blue
757 175.40 93.16 Green
758 164.81 80.83 Brown
759 173.27 78.43 Grey
760 169.88 91.69 Brown
761 164.23 78.95 Brown
762 173.65 63.78 Green
763 178.03 74.77 Grey
764 181.01 64.19 Blue
765 172.66 89.68 Green
766 167.56 77.80 Green
767 178.12 75.31 Green
768 166.72 80.59 Blue
769 182.65 82.26 Grey
770 171.72 74.41 Blue
771 166.09 76.62 Grey
772 181.88 73.97 Brown
773 167.16 91.88 Green
774 171.36 87.56 Brown
775 175.02 75.21 Green
776 169.55 104.21 Green
777 176.59 82.07 Blue
778 165.42 71.80 Green
779 165.28 83.93 Green
780 168.37 73.33 Grey
781 162.49 77.91 Brown
782 164.98 90.22 Grey
783 166.85 97.52 Grey
784 179.63 71.83 Grey
785 181.17 94.04 Brown
786 163.61 80.50 Blue
787 164.06 78.31 Blue
788 175.48 71.22 Green
789 169.71 75.33 Green
790 164.87 86.46 Blue
791 167.33 62.18 Brown
792 174.10 88.73 Green
793 163.84 86.79 Grey
794 171.01 88.12 Green
795 167.81 63.93 Brown
796 171.30 88.24 Brown
797 177.14 83.15 Green
798 166.00 58.66 Green
799 164.00 71.77 Blue
800 173.92 96.16 Grey
801 186.49 75.17 Green
802 177.72 76.34 Brown
803 168.64 81.95 Blue
804 165.53 78.08 Blue
805 176.76 80.04 Brown
806 161.24 84.16 Grey
807 173.18 68.94 Brown
808 171.78 79.08 Brown
809 171.78 71.89 Blue
810 173.57 72.97 Brown
811 167.95 93.72 Green
812 165.88 70.11 Blue
813 174.06 75.18 Green
814 172.18 97.41 Grey
815 160.13 83.10 Blue
816 171.47 73.90 Green
817 156.97 82.53 Brown
818 162.41 82.18 Blue
819 172.89 75.28 Blue
820 166.15 98.58 Grey
821 173.39 98.04 Grey
822 162.36 80.45 Grey
823 172.03 94.45 Blue
824 173.22 93.24 Green
825 174.20 81.11 Grey
826 176.17 92.28 Blue
827 159.19 93.72 Green
828 165.12 71.77 Grey
829 163.88 87.71 Grey
830 178.25 65.16 Blue
831 169.10 84.58 Brown
832 174.34 80.77 Grey
833 167.77 68.38 Green
834 163.67 78.18 Grey
835 174.88 65.89 Grey
836 167.80 85.85 Blue
837 166.86 73.36 Brown
838 179.33 75.85 Green
839 164.99 81.11 Blue
840 160.41 88.20 Blue
841 162.01 62.67 Green
842 171.02 84.10 Blue
843 167.13 69.47 Blue
844 165.07 90.08 Grey
845 169.01 58.07 Blue
846 171.05 81.41 Grey
847 164.38 84.70 Grey
848 175.75 88.35 Green
849 173.99 85.47 Blue
850 167.02 89.50 Brown
851 179.47 90.55 Green
852 180.05 57.27 Brown
853 171.43 84.60 Blue
854 169.34 56.95 Green
855 166.17 83.05 Brown
856 172.60 84.16 Blue
857 175.95 73.06 Green
858 184.34 88.29 Grey
859 164.43 52.49 Green
860 180.30 77.35 Brown
861 168.77 67.67 Grey
862 154.85 83.16 Brown
863 167.43 99.05 Green
864 153.55 71.55 Grey
865 158.52 85.64 Grey
866 166.01 69.37 Brown
867 164.43 76.99 Grey
868 175.45 68.77 Brown
869 184.21 68.52 Green
870 166.31 80.57 Green
871 171.12 81.33 Grey
872 163.77 76.05 Grey
873 175.04 77.48 Blue
874 176.59 102.74 Green
875 169.23 67.34 Green
876 165.06 84.15 Brown
877 169.10 72.25 Green
878 173.07 88.73 Brown
879 171.58 72.35 Brown
880 170.34 69.75 Brown
881 157.07 91.28 Green
882 174.68 86.03 Grey
883 169.55 105.16 Green
884 172.26 69.65 Brown
885 166.95 102.74 Blue
886 173.16 67.80 Grey
887 163.87 93.58 Green
888 164.05 53.66 Blue
889 175.62 83.99 Blue
890 175.04 90.40 Blue
891 175.00 69.03 Blue
892 163.12 83.92 Grey
893 160.41 81.10 Grey
894 164.59 76.25 Grey
895 170.98 81.28 Brown
896 176.30 64.79 Brown
897 174.58 77.13 Green
898 164.00 85.53 Brown
899 158.81 73.95 Brown
900 176.49 77.54 Blue
901 169.66 68.72 Grey
902 160.50 73.35 Brown
903 164.66 94.38 Green
904 170.40 90.15 Blue
905 163.80 96.31 Grey
906 168.98 78.93 Grey
907 179.11 84.44 Green
908 169.20 69.87 Green
909 170.75 84.63 Grey
910 171.31 84.22 Green
911 175.38 81.53 Grey
912 173.87 73.93 Grey
913 167.49 105.02 Blue
914 170.70 70.27 Green
915 167.82 77.02 Brown
916 172.47 90.07 Grey
917 166.75 89.11 Brown
918 180.36 66.44 Green
919 168.75 92.93 Grey
920 170.89 101.59 Brown
921 175.85 75.51 Grey
922 180.04 84.26 Blue
923 171.04 78.29 Blue
924 153.51 72.37 Brown
925 177.84 66.64 Brown
926 173.67 85.03 Blue
927 169.33 89.97 Grey
928 157.66 75.72 Green
929 168.23 88.40 Grey
930 163.20 112.01 Brown
931 166.49 84.73 Blue
932 164.94 55.95 Brown
933 184.23 77.38 Grey
934 163.89 57.70 Blue
935 177.42 88.59 Green
936 165.40 67.20 Brown
937 173.63 69.10 Brown
938 172.05 105.04 Grey
939 169.04 89.07 Grey
940 172.33 74.63 Green
941 163.11 81.14 Green
942 167.25 80.97 Brown
943 171.85 84.04 Brown
944 171.15 94.66 Blue
945 172.66 66.67 Grey
946 161.80 75.28 Blue
947 166.41 89.42 Green
948 170.99 80.73 Blue
949 173.01 87.67 Grey
950 171.35 66.75 Grey
951 160.63 91.08 Blue
952 174.08 66.82 Blue
953 172.22 89.78 Grey
954 172.57 90.79 Blue
955 172.31 75.52 Grey
956 178.45 66.69 Blue
957 168.67 84.34 Blue
958 162.35 68.54 Blue
959 168.17 58.81 Green
960 171.66 78.64 Green
961 172.03 69.68 Blue
962 176.08 85.25 Brown
963 175.58 64.86 Green
964 173.74 77.21 Grey
965 162.38 56.99 Green
966 177.40 67.66 Green
967 173.26 70.92 Blue
968 169.47 86.97 Grey
969 164.55 101.67 Green
970 171.09 95.36 Blue
971 168.34 79.49 Grey
972 173.36 75.20 Green
973 173.69 72.27 Grey
974 175.67 79.14 Grey
975 163.98 91.93 Green
976 170.40 89.87 Green
977 161.75 71.14 Brown
978 174.45 97.85 Brown
979 164.01 87.02 Brown
980 168.01 72.12 Green
981 166.01 70.70 Green
982 164.53 70.62 Grey
983 167.30 72.98 Brown
984 172.11 99.68 Green
985 172.45 84.12 Grey
986 169.16 78.10 Grey
987 168.53 87.21 Green
988 176.67 81.50 Green
989 169.90 95.50 Brown
990 171.53 77.40 Blue
991 171.34 72.32 Green
992 177.02 76.03 Grey
993 166.97 70.67 Blue
994 170.55 95.25 Blue
995 163.57 69.84 Brown
996 179.04 77.39 Blue
997 168.27 74.75 Green
998 178.57 96.92 Green
999 162.76 72.21 Grey
1000 164.82 80.71 Brown

Random assignment

I’m now going to flip (an imaginary, computer-generated) coin for each of these 1,000 individuals to assigned them to group A or B:

ID Height Weight Eye colour Group
1 169.67 66.25 Grey A
2 173.07 77.49 Blue B
3 177.22 84.55 Blue B
4 161.48 68.89 Blue B
5 182.28 79.94 Blue B
6 166.34 77.79 Grey A
7 174.67 77.06 Brown A
8 177.08 84.17 Brown B
9 165.39 67.19 Brown B
10 170.43 86.49 Blue B
11 173.90 81.97 Brown A
12 163.88 91.35 Blue B
13 170.08 90.36 Brown A
14 176.58 69.89 Grey B
15 163.34 84.29 Grey B
16 171.20 64.83 Green B
17 168.31 72.28 Blue A
18 173.78 85.94 Grey A
19 174.38 93.37 Grey B
20 167.46 60.09 Green B
21 179.75 80.55 Grey B
22 168.72 55.62 Grey B
23 163.97 69.88 Green A
24 167.42 64.65 Blue A
25 175.88 65.71 Grey B
26 169.16 70.00 Blue A
27 178.06 81.75 Green B
28 165.62 61.84 Green B
29 162.57 86.71 Green B
30 157.92 82.34 Green A
31 160.98 82.65 Green A
32 169.79 92.62 Grey A
33 163.95 93.35 Blue B
34 170.50 80.65 Green B
35 169.28 98.70 Green B
36 174.00 80.39 Green B
37 169.37 105.51 Brown A
38 163.08 79.24 Blue B
39 164.63 87.55 Blue A
40 173.05 70.55 Brown A
41 176.38 81.30 Blue A
42 173.28 76.12 Brown A
43 153.34 80.19 Brown B
44 176.71 88.35 Green B
45 165.33 64.82 Grey A
46 163.15 64.17 Grey A
47 166.44 80.59 Blue B
48 174.72 84.70 Grey B
49 163.70 76.83 Blue B
50 175.31 81.44 Brown B
51 156.73 89.48 Green B
52 167.73 80.08 Blue A
53 179.72 87.96 Blue A
54 169.92 85.19 Blue B
55 171.18 99.64 Brown B
56 155.76 88.90 Green A
57 162.26 56.96 Grey A
58 158.84 75.51 Grey A
59 150.74 70.19 Green A
60 176.27 70.51 Green B
61 169.75 75.31 Blue A
62 175.34 89.45 Grey A
63 169.99 97.23 Green A
64 164.14 87.03 Green B
65 159.63 75.01 Blue A
66 161.95 58.41 Green A
67 164.31 93.66 Grey A
68 153.77 80.86 Brown A
69 171.99 89.15 Grey B
70 167.41 89.92 Brown B
71 165.72 90.26 Blue B
72 166.25 78.84 Grey B
73 183.10 66.15 Blue A
74 167.66 79.84 Grey A
75 171.15 86.23 Blue A
76 167.40 71.65 Grey A
77 169.46 90.00 Grey A
78 155.28 77.01 Blue B
79 163.38 59.35 Grey B
80 178.22 72.58 Green B
81 161.95 77.90 Grey A
82 177.46 73.13 Grey B
83 170.53 69.19 Green B
84 168.86 89.00 Green A
85 179.85 85.48 Grey A
86 171.53 77.63 Green A
87 175.24 75.91 Blue A
88 181.81 94.81 Blue B
89 164.16 79.06 Brown A
90 160.76 89.50 Brown B
91 161.75 89.30 Blue A
92 168.40 80.54 Green A
93 164.69 90.93 Blue A
94 164.61 81.30 Blue A
95 168.25 69.42 Grey A
96 172.41 68.56 Blue A
97 170.50 82.31 Blue A
98 177.90 66.75 Green A
99 170.18 91.41 Green A
100 176.35 88.30 Grey B
101 163.61 79.84 Grey B
102 170.37 83.49 Blue A
103 162.80 83.93 Grey B
104 165.08 100.79 Green B
105 168.06 77.16 Grey A
106 178.30 87.79 Blue B
107 158.56 74.17 Brown B
108 176.90 90.69 Grey B
109 165.05 83.44 Brown A
110 176.26 88.39 Brown A
111 164.35 94.24 Brown B
112 162.52 89.67 Brown B
113 163.37 82.60 Brown A
114 168.13 85.49 Brown B
115 167.09 82.06 Blue A
116 166.98 75.99 Green B
117 170.33 80.69 Brown A
118 168.70 79.16 Grey A
119 171.32 66.15 Blue A
120 178.29 79.79 Blue B
121 176.63 93.71 Grey B
122 170.27 85.53 Grey B
123 168.90 68.24 Brown B
124 174.65 54.62 Green A
125 173.43 71.15 Brown B
126 163.25 84.85 Blue B
127 173.17 74.42 Blue A
128 159.67 69.37 Brown B
129 170.33 88.54 Brown A
130 176.08 70.47 Blue A
131 165.73 87.17 Brown B
132 175.71 64.11 Brown A
133 158.95 90.90 Brown A
134 172.04 94.32 Brown B
135 176.93 82.80 Grey B
136 166.87 94.96 Brown B
137 168.25 83.90 Green B
138 166.54 76.47 Grey A
139 178.04 75.91 Green A
140 171.63 85.71 Brown B
141 174.82 87.38 Brown B
142 170.00 69.44 Blue B
143 165.06 91.37 Blue A
144 163.98 91.13 Brown B
145 170.54 62.06 Blue A
146 164.56 63.25 Grey B
147 165.24 83.45 Brown A
148 169.56 76.95 Green A
149 163.94 74.58 Brown A
150 174.57 70.56 Grey B
151 172.37 65.14 Brown A
152 168.59 75.45 Grey A
153 178.13 69.47 Brown A
154 158.96 80.51 Grey B
155 175.68 93.20 Grey B
156 165.04 99.50 Brown A
157 168.45 46.30 Brown B
158 167.92 77.50 Grey A
159 161.46 95.20 Brown B
160 168.03 66.89 Blue A
161 160.48 77.25 Brown A
162 179.46 78.11 Brown B
163 166.45 68.23 Brown B
164 169.67 77.34 Green B
165 177.31 89.58 Green A
166 171.62 78.98 Brown B
167 170.61 65.36 Green B
168 166.52 69.72 Blue A
169 167.49 62.28 Green B
170 167.58 82.87 Blue A
171 172.18 103.63 Grey A
172 152.55 64.18 Brown B
173 175.02 91.32 Blue A
174 171.82 64.80 Blue B
175 163.88 73.52 Green B
176 169.69 80.78 Grey A
177 166.43 75.20 Brown B
178 174.37 72.38 Blue B
179 173.27 72.09 Green A
180 162.33 71.77 Green B
181 167.34 72.77 Green B
182 172.31 94.33 Grey B
183 157.00 78.63 Brown B
184 173.55 62.94 Brown A
185 171.23 80.66 Grey B
186 167.64 86.47 Blue A
187 171.14 83.63 Grey A
188 171.20 81.59 Green B
189 171.83 86.94 Blue B
190 174.15 74.26 Green B
191 170.33 81.00 Grey B
192 177.74 89.21 Brown B
193 171.34 96.63 Blue A
194 158.77 90.77 Green B
195 173.11 77.73 Green B
196 173.39 83.76 Grey B
197 173.02 59.01 Blue A
198 163.17 78.44 Grey B
199 173.85 91.39 Blue B
200 168.48 101.98 Brown B
201 176.13 103.48 Green B
202 162.57 91.50 Green B
203 173.96 76.52 Grey A
204 165.54 61.81 Grey A
205 166.79 92.50 Brown B
206 167.15 90.50 Green B
207 170.88 72.43 Brown A
208 176.89 70.32 Grey A
209 173.79 83.16 Brown B
210 161.67 93.15 Blue B
211 180.41 62.05 Blue B
212 168.30 75.67 Grey A
213 169.95 71.30 Green B
214 170.10 79.64 Green B
215 169.62 74.89 Blue A
216 165.53 72.20 Blue B
217 174.74 76.32 Blue A
218 165.04 58.08 Brown A
219 169.19 93.89 Brown B
220 171.67 83.03 Grey A
221 180.26 93.00 Brown B
222 173.77 88.11 Brown B
223 171.43 79.40 Brown A
224 165.79 79.97 Green B
225 175.11 83.71 Green B
226 159.92 91.34 Green B
227 173.42 94.30 Blue B
228 174.38 91.77 Blue A
229 167.75 83.27 Blue A
230 173.25 65.85 Blue A
231 166.84 74.42 Grey B
232 166.09 60.93 Green B
233 170.41 91.47 Green A
234 177.48 82.41 Grey B
235 168.34 78.32 Grey B
236 179.02 70.87 Brown B
237 172.50 58.46 Green B
238 164.71 79.70 Green B
239 167.53 78.63 Blue A
240 165.53 76.99 Brown A
241 160.51 76.17 Blue B
242 170.72 71.19 Blue B
243 170.25 96.24 Grey B
244 168.02 76.44 Brown A
245 160.35 66.39 Blue B
246 168.02 75.83 Brown B
247 174.09 85.73 Grey A
248 180.22 86.70 Green A
249 178.29 86.30 Blue A
250 168.96 86.36 Blue B
251 171.80 87.82 Green A
252 166.88 83.18 Blue A
253 163.95 67.66 Green B
254 165.55 51.50 Grey A
255 165.44 73.29 Grey A
256 168.18 82.85 Grey B
257 173.37 104.05 Grey A
258 171.92 91.95 Blue A
259 177.11 65.89 Grey A
260 164.24 85.79 Brown A
261 162.73 77.69 Grey B
262 171.02 66.53 Green A
263 171.90 69.30 Green A
264 178.69 82.23 Blue B
265 171.34 89.01 Blue B
266 162.81 74.98 Brown B
267 164.44 92.84 Blue B
268 176.29 75.55 Grey A
269 179.40 77.74 Blue A
270 167.72 76.49 Grey B
271 169.35 79.56 Green B
272 167.85 74.17 Blue B
273 171.22 66.68 Grey B
274 174.01 73.62 Blue B
275 165.79 92.47 Green A
276 170.44 84.38 Brown B
277 171.79 75.18 Blue A
278 167.62 76.82 Green A
279 170.43 97.69 Grey A
280 169.49 94.31 Brown B
281 168.82 80.38 Green A
282 163.19 74.14 Brown B
283 172.74 64.72 Grey B
284 172.99 85.37 Blue B
285 181.01 64.75 Green B
286 167.32 70.22 Green A
287 161.76 79.12 Green A
288 175.39 65.83 Blue B
289 172.49 58.91 Grey B
290 169.48 77.32 Brown A
291 171.04 63.96 Blue B
292 166.91 69.87 Green A
293 172.16 72.75 Green A
294 163.50 74.85 Brown B
295 169.25 71.03 Blue A
296 156.55 85.45 Green B
297 176.93 89.49 Brown A
298 172.39 74.21 Blue B
299 175.32 80.00 Grey A
300 173.70 70.07 Grey B
301 162.02 94.04 Grey B
302 165.75 65.25 Blue B
303 157.00 85.62 Blue A
304 176.94 77.30 Blue B
305 183.07 80.75 Grey A
306 164.46 63.66 Brown A
307 170.59 74.69 Brown A
308 174.93 83.28 Grey B
309 167.43 58.90 Blue A
310 178.23 77.13 Blue B
311 173.08 82.83 Brown A
312 177.13 72.76 Grey B
313 165.98 72.69 Blue B
314 159.16 82.02 Green B
315 180.21 80.08 Grey A
316 179.47 84.34 Grey A
317 169.05 87.30 Grey A
318 167.53 69.95 Grey B
319 165.03 85.89 Green B
320 178.33 84.27 Brown A
321 163.15 57.84 Blue B
322 174.94 69.65 Green B
323 169.97 80.56 Brown B
324 169.57 91.61 Brown B
325 163.56 78.59 Brown B
326 166.60 72.65 Green A
327 167.92 79.59 Brown A
328 165.29 77.95 Green A
329 163.50 82.21 Brown A
330 167.69 80.23 Brown A
331 167.34 81.09 Brown B
332 183.18 73.28 Grey B
333 168.05 63.28 Brown B
334 168.46 83.19 Brown B
335 169.69 76.85 Brown A
336 162.30 74.61 Brown B
337 161.20 62.75 Brown A
338 175.36 71.88 Green B
339 175.72 60.90 Blue A
340 169.14 85.45 Brown A
341 171.49 76.97 Grey B
342 165.65 85.93 Blue B
343 170.67 76.84 Grey A
344 161.02 90.57 Brown A
345 172.42 95.07 Green A
346 167.73 76.67 Grey B
347 166.65 84.11 Green A
348 185.80 74.56 Green B
349 176.11 84.55 Brown A
350 182.05 65.14 Blue A
351 158.28 94.00 Blue A
352 168.25 86.08 Blue B
353 175.35 84.48 Brown A
354 167.74 90.33 Blue A
355 176.02 78.57 Blue B
356 174.01 65.81 Blue B
357 170.95 72.78 Brown B
358 174.90 74.07 Blue A
359 174.78 69.25 Green A
360 180.85 78.87 Blue A
361 171.75 74.72 Blue A
362 170.16 63.45 Green B
363 163.83 74.78 Brown B
364 163.54 66.85 Green A
365 166.24 63.27 Blue B
366 168.59 75.87 Blue A
367 175.89 88.61 Grey B
368 162.23 83.82 Brown A
369 163.71 89.80 Blue A
370 169.93 78.73 Grey A
371 177.18 90.98 Grey B
372 170.24 75.03 Blue B
373 174.87 75.69 Green A
374 162.40 79.13 Green B
375 170.13 79.79 Grey B
376 166.94 74.00 Green A
377 168.56 80.46 Green A
378 174.01 88.67 Blue A
379 170.75 69.65 Blue A
380 164.57 68.81 Grey A
381 166.38 83.07 Brown A
382 173.81 103.64 Grey A
383 176.74 83.21 Blue A
384 172.77 86.78 Grey A
385 167.69 83.88 Brown B
386 169.92 64.74 Green B
387 164.29 76.11 Brown A
388 169.80 69.92 Blue A
389 157.95 68.20 Blue B
390 170.37 77.17 Brown A
391 166.87 83.42 Blue A
392 176.80 73.38 Blue B
393 174.20 86.58 Brown A
394 171.09 79.50 Grey B
395 180.39 90.74 Blue A
396 162.37 94.78 Brown A
397 167.97 82.86 Grey A
398 173.60 75.71 Brown B
399 163.64 76.49 Grey A
400 168.62 56.65 Brown B
401 161.10 68.50 Brown A
402 174.01 91.69 Grey A
403 172.16 79.52 Green B
404 177.43 81.52 Green A
405 177.17 78.78 Blue A
406 168.58 82.51 Grey B
407 162.25 84.93 Blue B
408 167.03 79.73 Green B
409 171.97 75.60 Brown A
410 165.83 70.53 Grey B
411 155.46 80.38 Blue A
412 161.78 83.76 Green B
413 175.19 70.08 Blue A
414 163.85 79.53 Brown B
415 177.12 92.35 Brown B
416 175.46 79.74 Grey A
417 167.62 92.91 Green B
418 176.03 92.59 Brown A
419 163.96 80.21 Blue A
420 168.65 89.09 Brown A
421 172.76 78.41 Brown A
422 171.72 93.61 Brown A
423 170.68 76.29 Brown B
424 176.70 82.33 Green B
425 161.56 82.09 Brown B
426 175.66 84.89 Grey A
427 161.58 106.96 Grey B
428 155.74 85.07 Green B
429 168.39 80.94 Green A
430 162.14 88.56 Blue B
431 175.06 65.90 Grey B
432 174.33 84.15 Blue B
433 171.29 75.52 Blue A
434 166.95 93.32 Blue B
435 164.80 71.76 Green B
436 166.99 85.78 Grey A
437 176.20 92.28 Green B
438 174.71 71.88 Brown A
439 168.13 68.53 Grey B
440 169.54 83.67 Brown B
441 169.34 91.25 Grey A
442 182.19 68.10 Brown B
443 168.11 61.86 Grey A
444 174.34 79.88 Blue A
445 171.55 90.24 Green A
446 169.41 72.22 Brown B
447 172.31 86.63 Grey A
448 160.99 71.93 Green B
449 168.53 58.73 Brown A
450 168.18 73.21 Grey A
451 174.76 91.14 Grey A
452 178.12 83.63 Blue B
453 168.93 70.12 Green B
454 170.28 92.61 Green B
455 169.80 78.81 Brown A
456 174.86 94.98 Grey B
457 175.70 83.20 Blue B
458 172.81 79.91 Blue B
459 163.41 67.28 Green A
460 158.94 67.60 Blue B
461 170.82 93.56 Blue A
462 164.87 88.40 Green A
463 163.52 73.19 Blue A
464 171.90 74.78 Green A
465 166.24 93.86 Brown A
466 171.65 83.36 Brown A
467 175.13 74.10 Grey B
468 166.62 70.00 Green A
469 170.90 91.01 Green B
470 176.05 78.23 Blue B
471 171.29 77.75 Blue B
472 172.61 73.95 Blue A
473 178.86 72.93 Grey B
474 172.50 84.50 Blue B
475 163.77 85.30 Grey B
476 166.20 79.78 Green A
477 172.29 70.87 Brown B
478 175.71 80.55 Blue A
479 166.11 73.41 Green A
480 168.12 87.81 Blue A
481 167.60 79.10 Green B
482 166.16 80.94 Blue B
483 173.79 86.86 Brown B
484 174.91 71.13 Brown B
485 161.76 83.33 Brown A
486 169.21 71.44 Brown B
487 176.63 90.14 Blue A
488 167.60 81.52 Brown B
489 167.94 82.32 Blue B
490 175.36 97.69 Blue B
491 170.48 97.11 Green A
492 159.35 77.57 Blue B
493 172.98 87.22 Blue A
494 167.08 88.95 Grey A
495 183.28 84.62 Brown B
496 171.71 72.31 Brown B
497 175.19 82.13 Grey A
498 165.91 74.58 Grey A
499 172.28 77.61 Green B
500 158.19 82.44 Blue B
501 171.93 77.86 Grey B
502 165.49 67.09 Brown B
503 172.39 88.61 Grey B
504 156.17 66.08 Green A
505 165.22 77.81 Green B
506 168.80 60.26 Grey A
507 163.78 68.12 Grey B
508 169.06 67.72 Blue B
509 161.53 61.52 Brown B
510 167.96 77.25 Blue A
511 163.69 87.84 Grey A
512 167.75 66.31 Blue A
513 176.16 86.21 Brown B
514 177.91 89.78 Brown B
515 156.78 78.65 Grey A
516 167.26 64.35 Brown A
517 179.76 76.17 Grey B
518 165.62 78.30 Blue A
519 162.94 70.85 Blue A
520 166.69 74.72 Green B
521 173.34 89.88 Blue B
522 168.59 89.67 Blue B
523 176.38 88.78 Brown B
524 166.84 79.39 Blue B
525 163.21 94.70 Blue A
526 165.76 81.70 Grey B
527 170.15 95.34 Green A
528 171.57 80.29 Brown A
529 164.89 82.41 Blue B
530 169.42 84.39 Brown A
531 178.41 79.31 Grey A
532 165.07 75.72 Green A
533 162.92 83.84 Blue A
534 174.29 83.60 Brown B
535 163.15 76.57 Grey A
536 165.51 79.69 Blue B
537 170.40 86.27 Blue A
538 170.53 73.89 Green A
539 169.18 69.30 Brown A
540 164.75 78.19 Grey A
541 173.57 71.87 Grey A
542 170.77 69.52 Grey A
543 160.79 90.26 Blue B
544 172.72 56.69 Green A
545 163.43 100.84 Grey B
546 171.23 88.51 Grey A
547 174.08 80.74 Green A
548 166.88 60.29 Green A
549 178.43 77.01 Blue A
550 166.30 73.07 Grey B
551 167.15 77.73 Green A
552 181.67 81.12 Brown B
553 177.56 89.42 Green A
554 162.68 90.49 Grey B
555 177.65 74.99 Grey A
556 167.79 63.24 Blue A
557 175.02 89.23 Green B
558 173.32 69.26 Green B
559 164.91 71.33 Brown B
560 174.35 91.37 Green B
561 169.56 60.45 Grey A
562 163.08 51.92 Green A
563 167.36 86.34 Blue A
564 176.27 65.28 Grey A
565 160.89 67.87 Brown A
566 181.94 72.99 Blue A
567 167.86 79.87 Grey B
568 171.58 76.01 Brown A
569 182.72 78.96 Blue A
570 166.58 82.91 Brown A
571 172.64 68.26 Brown A
572 168.52 76.11 Brown A
573 177.49 87.15 Grey B
574 179.59 63.62 Grey B
575 176.72 69.82 Grey A
576 162.78 60.89 Blue A
577 177.53 78.34 Brown B
578 170.44 81.64 Green B
579 168.92 59.25 Green B
580 169.79 85.94 Green B
581 171.44 69.18 Grey A
582 164.51 78.68 Green A
583 168.27 76.40 Green A
584 168.93 91.60 Green A
585 166.47 82.87 Grey A
586 160.76 65.36 Blue B
587 169.52 75.73 Green A
588 176.90 89.42 Grey A
589 174.21 94.73 Green B
590 187.87 68.18 Green A
591 166.11 76.04 Green A
592 176.90 48.41 Blue B
593 170.32 79.63 Grey B
594 160.28 73.36 Blue A
595 180.06 83.45 Green A
596 186.73 83.70 Green A
597 167.46 86.99 Grey A
598 170.29 89.85 Brown B
599 160.97 76.77 Grey A
600 173.71 74.82 Brown B
601 167.63 86.52 Grey B
602 169.09 66.55 Blue A
603 159.16 88.91 Blue B
604 168.37 91.49 Green B
605 176.00 70.23 Blue B
606 156.01 69.02 Brown A
607 166.90 66.61 Grey B
608 165.36 74.44 Brown B
609 166.17 62.86 Green A
610 176.70 87.29 Blue B
611 177.42 78.15 Brown A
612 164.72 99.82 Grey A
613 162.33 70.30 Green A
614 167.08 75.96 Blue B
615 172.92 89.45 Blue A
616 176.38 90.26 Brown B
617 164.35 99.98 Green A
618 161.33 89.42 Green A
619 172.89 72.92 Blue A
620 170.09 67.26 Blue A
621 162.07 85.68 Blue B
622 166.42 79.14 Green A
623 164.99 86.53 Blue B
624 174.66 97.71 Brown B
625 171.69 83.08 Blue A
626 181.11 82.09 Grey A
627 174.95 90.11 Blue A
628 158.99 77.31 Brown A
629 158.09 82.81 Brown A
630 174.77 97.32 Grey A
631 169.11 93.13 Brown A
632 172.59 93.14 Brown A
633 169.70 79.31 Blue B
634 167.12 82.27 Green B
635 167.64 92.75 Green B
636 174.27 72.74 Grey A
637 172.07 75.33 Blue A
638 164.73 79.23 Brown B
639 173.21 66.91 Brown B
640 171.36 90.01 Green B
641 181.71 76.30 Grey A
642 182.42 93.69 Blue B
643 169.61 77.74 Green A
644 169.94 80.28 Green A
645 161.31 85.89 Brown B
646 170.82 69.60 Grey A
647 167.34 72.56 Grey B
648 173.51 77.10 Grey B
649 177.17 82.81 Grey B
650 173.15 92.59 Green B
651 181.22 85.25 Blue A
652 162.52 74.60 Green A
653 168.46 83.11 Brown B
654 174.55 88.56 Brown B
655 169.81 60.38 Grey A
656 168.51 86.75 Brown B
657 166.28 73.65 Brown A
658 165.82 74.41 Grey A
659 169.34 88.84 Blue B
660 168.72 69.02 Brown B
661 171.23 71.29 Green A
662 153.97 84.77 Grey B
663 166.65 88.47 Green A
664 172.50 91.36 Blue B
665 169.29 81.93 Grey B
666 168.82 75.03 Grey A
667 165.97 76.03 Brown B
668 185.29 79.42 Brown B
669 172.73 79.34 Brown B
670 168.55 95.58 Blue B
671 166.32 80.88 Grey B
672 169.42 94.68 Grey B
673 175.61 90.74 Blue B
674 162.14 81.25 Green A
675 168.29 85.46 Grey B
676 174.46 81.54 Brown A
677 172.01 66.11 Grey B
678 173.39 82.46 Brown B
679 163.49 86.31 Brown A
680 173.13 86.92 Brown B
681 175.27 78.20 Green B
682 172.68 68.31 Grey B
683 172.69 84.18 Blue B
684 174.48 70.18 Brown B
685 171.44 87.02 Brown B
686 180.49 66.54 Blue B
687 170.42 88.85 Blue B
688 174.09 79.16 Grey B
689 170.19 82.46 Brown B
690 173.29 77.31 Green B
691 177.17 72.70 Green B
692 184.40 88.05 Grey A
693 164.17 82.73 Green B
694 176.87 50.04 Grey B
695 163.74 77.81 Green A
696 167.06 85.49 Green A
697 166.98 92.84 Grey A
698 178.55 73.59 Blue A
699 167.90 100.46 Grey B
700 172.32 75.31 Green B
701 175.94 85.47 Green A
702 174.88 73.51 Green B
703 165.82 70.07 Brown A
704 166.91 86.90 Brown A
705 165.89 75.70 Blue B
706 178.13 74.46 Grey A
707 169.14 84.96 Green A
708 171.26 93.53 Grey A
709 179.73 88.67 Grey B
710 174.96 69.16 Green A
711 171.94 66.08 Grey A
712 171.25 76.27 Brown A
713 168.01 80.72 Green B
714 170.76 102.41 Grey B
715 172.41 98.88 Brown A
716 164.50 81.08 Green B
717 184.19 85.59 Grey B
718 174.84 96.25 Grey A
719 175.76 84.99 Blue A
720 174.04 83.28 Blue B
721 169.56 63.95 Grey B
722 174.79 64.56 Brown A
723 170.61 90.52 Green A
724 167.94 75.20 Blue A
725 166.65 83.80 Blue A
726 166.59 87.99 Brown A
727 162.11 70.68 Brown A
728 162.95 71.83 Blue B
729 174.61 70.39 Brown A
730 169.53 80.73 Green B
731 164.74 74.89 Brown A
732 167.52 65.03 Grey A
733 179.17 79.83 Brown B
734 164.45 84.92 Blue B
735 176.57 82.35 Grey B
736 179.69 53.65 Brown B
737 178.80 70.41 Green A
738 169.55 95.39 Blue B
739 171.04 90.30 Grey B
740 159.18 84.80 Blue B
741 167.39 97.14 Blue A
742 171.59 94.06 Green A
743 165.80 70.18 Blue B
744 180.39 83.52 Blue A
745 172.28 84.92 Grey B
746 162.85 84.10 Grey B
747 166.76 67.13 Green A
748 172.28 70.42 Grey B
749 167.07 74.62 Brown A
750 177.60 76.73 Blue B
751 166.38 81.71 Green B
752 173.11 86.63 Green B
753 168.69 72.60 Grey B
754 182.63 92.33 Green A
755 170.80 69.58 Grey A
756 162.67 94.45 Blue A
757 175.40 93.16 Green A
758 164.81 80.83 Brown B
759 173.27 78.43 Grey A
760 169.88 91.69 Brown A
761 164.23 78.95 Brown B
762 173.65 63.78 Green A
763 178.03 74.77 Grey A
764 181.01 64.19 Blue B
765 172.66 89.68 Green A
766 167.56 77.80 Green A
767 178.12 75.31 Green B
768 166.72 80.59 Blue B
769 182.65 82.26 Grey A
770 171.72 74.41 Blue B
771 166.09 76.62 Grey B
772 181.88 73.97 Brown B
773 167.16 91.88 Green A
774 171.36 87.56 Brown A
775 175.02 75.21 Green A
776 169.55 104.21 Green A
777 176.59 82.07 Blue A
778 165.42 71.80 Green B
779 165.28 83.93 Green A
780 168.37 73.33 Grey A
781 162.49 77.91 Brown A
782 164.98 90.22 Grey B
783 166.85 97.52 Grey B
784 179.63 71.83 Grey A
785 181.17 94.04 Brown B
786 163.61 80.50 Blue A
787 164.06 78.31 Blue B
788 175.48 71.22 Green A
789 169.71 75.33 Green B
790 164.87 86.46 Blue B
791 167.33 62.18 Brown B
792 174.10 88.73 Green A
793 163.84 86.79 Grey A
794 171.01 88.12 Green A
795 167.81 63.93 Brown A
796 171.30 88.24 Brown A
797 177.14 83.15 Green B
798 166.00 58.66 Green A
799 164.00 71.77 Blue A
800 173.92 96.16 Grey B
801 186.49 75.17 Green A
802 177.72 76.34 Brown B
803 168.64 81.95 Blue B
804 165.53 78.08 Blue A
805 176.76 80.04 Brown A
806 161.24 84.16 Grey A
807 173.18 68.94 Brown B
808 171.78 79.08 Brown A
809 171.78 71.89 Blue A
810 173.57 72.97 Brown B
811 167.95 93.72 Green B
812 165.88 70.11 Blue B
813 174.06 75.18 Green A
814 172.18 97.41 Grey B
815 160.13 83.10 Blue A
816 171.47 73.90 Green A
817 156.97 82.53 Brown A
818 162.41 82.18 Blue B
819 172.89 75.28 Blue A
820 166.15 98.58 Grey B
821 173.39 98.04 Grey B
822 162.36 80.45 Grey A
823 172.03 94.45 Blue A
824 173.22 93.24 Green A
825 174.20 81.11 Grey B
826 176.17 92.28 Blue A
827 159.19 93.72 Green B
828 165.12 71.77 Grey A
829 163.88 87.71 Grey A
830 178.25 65.16 Blue A
831 169.10 84.58 Brown A
832 174.34 80.77 Grey A
833 167.77 68.38 Green A
834 163.67 78.18 Grey B
835 174.88 65.89 Grey B
836 167.80 85.85 Blue B
837 166.86 73.36 Brown A
838 179.33 75.85 Green A
839 164.99 81.11 Blue B
840 160.41 88.20 Blue A
841 162.01 62.67 Green B
842 171.02 84.10 Blue A
843 167.13 69.47 Blue B
844 165.07 90.08 Grey B
845 169.01 58.07 Blue B
846 171.05 81.41 Grey A
847 164.38 84.70 Grey B
848 175.75 88.35 Green A
849 173.99 85.47 Blue A
850 167.02 89.50 Brown B
851 179.47 90.55 Green A
852 180.05 57.27 Brown B
853 171.43 84.60 Blue A
854 169.34 56.95 Green A
855 166.17 83.05 Brown B
856 172.60 84.16 Blue A
857 175.95 73.06 Green A
858 184.34 88.29 Grey A
859 164.43 52.49 Green A
860 180.30 77.35 Brown A
861 168.77 67.67 Grey B
862 154.85 83.16 Brown B
863 167.43 99.05 Green A
864 153.55 71.55 Grey B
865 158.52 85.64 Grey B
866 166.01 69.37 Brown B
867 164.43 76.99 Grey A
868 175.45 68.77 Brown B
869 184.21 68.52 Green B
870 166.31 80.57 Green A
871 171.12 81.33 Grey A
872 163.77 76.05 Grey B
873 175.04 77.48 Blue A
874 176.59 102.74 Green A
875 169.23 67.34 Green A
876 165.06 84.15 Brown B
877 169.10 72.25 Green B
878 173.07 88.73 Brown A
879 171.58 72.35 Brown A
880 170.34 69.75 Brown B
881 157.07 91.28 Green B
882 174.68 86.03 Grey A
883 169.55 105.16 Green A
884 172.26 69.65 Brown A
885 166.95 102.74 Blue B
886 173.16 67.80 Grey A
887 163.87 93.58 Green B
888 164.05 53.66 Blue A
889 175.62 83.99 Blue B
890 175.04 90.40 Blue A
891 175.00 69.03 Blue A
892 163.12 83.92 Grey B
893 160.41 81.10 Grey B
894 164.59 76.25 Grey B
895 170.98 81.28 Brown A
896 176.30 64.79 Brown A
897 174.58 77.13 Green B
898 164.00 85.53 Brown B
899 158.81 73.95 Brown A
900 176.49 77.54 Blue A
901 169.66 68.72 Grey B
902 160.50 73.35 Brown A
903 164.66 94.38 Green A
904 170.40 90.15 Blue B
905 163.80 96.31 Grey A
906 168.98 78.93 Grey A
907 179.11 84.44 Green B
908 169.20 69.87 Green A
909 170.75 84.63 Grey A
910 171.31 84.22 Green B
911 175.38 81.53 Grey A
912 173.87 73.93 Grey B
913 167.49 105.02 Blue A
914 170.70 70.27 Green B
915 167.82 77.02 Brown B
916 172.47 90.07 Grey A
917 166.75 89.11 Brown A
918 180.36 66.44 Green B
919 168.75 92.93 Grey B
920 170.89 101.59 Brown B
921 175.85 75.51 Grey B
922 180.04 84.26 Blue A
923 171.04 78.29 Blue B
924 153.51 72.37 Brown A
925 177.84 66.64 Brown B
926 173.67 85.03 Blue B
927 169.33 89.97 Grey A
928 157.66 75.72 Green B
929 168.23 88.40 Grey B
930 163.20 112.01 Brown B
931 166.49 84.73 Blue A
932 164.94 55.95 Brown A
933 184.23 77.38 Grey A
934 163.89 57.70 Blue B
935 177.42 88.59 Green B
936 165.40 67.20 Brown B
937 173.63 69.10 Brown A
938 172.05 105.04 Grey A
939 169.04 89.07 Grey A
940 172.33 74.63 Green A
941 163.11 81.14 Green B
942 167.25 80.97 Brown B
943 171.85 84.04 Brown B
944 171.15 94.66 Blue B
945 172.66 66.67 Grey A
946 161.80 75.28 Blue A
947 166.41 89.42 Green A
948 170.99 80.73 Blue B
949 173.01 87.67 Grey B
950 171.35 66.75 Grey A
951 160.63 91.08 Blue A
952 174.08 66.82 Blue B
953 172.22 89.78 Grey A
954 172.57 90.79 Blue B
955 172.31 75.52 Grey A
956 178.45 66.69 Blue A
957 168.67 84.34 Blue A
958 162.35 68.54 Blue A
959 168.17 58.81 Green B
960 171.66 78.64 Green B
961 172.03 69.68 Blue A
962 176.08 85.25 Brown A
963 175.58 64.86 Green B
964 173.74 77.21 Grey A
965 162.38 56.99 Green B
966 177.40 67.66 Green A
967 173.26 70.92 Blue A
968 169.47 86.97 Grey B
969 164.55 101.67 Green B
970 171.09 95.36 Blue B
971 168.34 79.49 Grey B
972 173.36 75.20 Green B
973 173.69 72.27 Grey A
974 175.67 79.14 Grey A
975 163.98 91.93 Green A
976 170.40 89.87 Green B
977 161.75 71.14 Brown A
978 174.45 97.85 Brown A
979 164.01 87.02 Brown A
980 168.01 72.12 Green A
981 166.01 70.70 Green A
982 164.53 70.62 Grey A
983 167.30 72.98 Brown A
984 172.11 99.68 Green A
985 172.45 84.12 Grey A
986 169.16 78.10 Grey B
987 168.53 87.21 Green A
988 176.67 81.50 Green A
989 169.90 95.50 Brown B
990 171.53 77.40 Blue A
991 171.34 72.32 Green A
992 177.02 76.03 Grey A
993 166.97 70.67 Blue B
994 170.55 95.25 Blue A
995 163.57 69.84 Brown B
996 179.04 77.39 Blue B
997 168.27 74.75 Green B
998 178.57 96.92 Green B
999 162.76 72.21 Grey A
1000 164.82 80.71 Brown B

How similar are these groups?

Let’s first check their heights:

How similar are these groups?

And their weights:

How similar are these groups?

And their eye colors:

Making sure this wasn’t a fluke

Let’s re-run this:

ID Height Weight Eye colour Group
1 169.67 66.25 Grey A
2 173.07 77.49 Blue A
3 177.22 84.55 Blue B
4 161.48 68.89 Blue B
5 182.28 79.94 Blue B
6 166.34 77.79 Grey A
7 174.67 77.06 Brown B
8 177.08 84.17 Brown A
9 165.39 67.19 Brown B
10 170.43 86.49 Blue A
11 173.90 81.97 Brown B
12 163.88 91.35 Blue A
13 170.08 90.36 Brown A
14 176.58 69.89 Grey B
15 163.34 84.29 Grey B
16 171.20 64.83 Green B
17 168.31 72.28 Blue B
18 173.78 85.94 Grey B
19 174.38 93.37 Grey A
20 167.46 60.09 Green B
21 179.75 80.55 Grey A
22 168.72 55.62 Grey A
23 163.97 69.88 Green A
24 167.42 64.65 Blue B
25 175.88 65.71 Grey A
26 169.16 70.00 Blue A
27 178.06 81.75 Green A
28 165.62 61.84 Green B
29 162.57 86.71 Green B
30 157.92 82.34 Green B
31 160.98 82.65 Green B
32 169.79 92.62 Grey A
33 163.95 93.35 Blue A
34 170.50 80.65 Green B
35 169.28 98.70 Green A
36 174.00 80.39 Green B
37 169.37 105.51 Brown B
38 163.08 79.24 Blue A
39 164.63 87.55 Blue A
40 173.05 70.55 Brown B
41 176.38 81.30 Blue B
42 173.28 76.12 Brown A
43 153.34 80.19 Brown B
44 176.71 88.35 Green A
45 165.33 64.82 Grey B
46 163.15 64.17 Grey B
47 166.44 80.59 Blue A
48 174.72 84.70 Grey A
49 163.70 76.83 Blue B
50 175.31 81.44 Brown A
51 156.73 89.48 Green A
52 167.73 80.08 Blue B
53 179.72 87.96 Blue A
54 169.92 85.19 Blue B
55 171.18 99.64 Brown A
56 155.76 88.90 Green B
57 162.26 56.96 Grey A
58 158.84 75.51 Grey A
59 150.74 70.19 Green B
60 176.27 70.51 Green B
61 169.75 75.31 Blue A
62 175.34 89.45 Grey A
63 169.99 97.23 Green B
64 164.14 87.03 Green A
65 159.63 75.01 Blue B
66 161.95 58.41 Green B
67 164.31 93.66 Grey B
68 153.77 80.86 Brown B
69 171.99 89.15 Grey B
70 167.41 89.92 Brown A
71 165.72 90.26 Blue B
72 166.25 78.84 Grey A
73 183.10 66.15 Blue A
74 167.66 79.84 Grey A
75 171.15 86.23 Blue B
76 167.40 71.65 Grey B
77 169.46 90.00 Grey A
78 155.28 77.01 Blue B
79 163.38 59.35 Grey B
80 178.22 72.58 Green B
81 161.95 77.90 Grey A
82 177.46 73.13 Grey A
83 170.53 69.19 Green B
84 168.86 89.00 Green B
85 179.85 85.48 Grey B
86 171.53 77.63 Green B
87 175.24 75.91 Blue B
88 181.81 94.81 Blue B
89 164.16 79.06 Brown A
90 160.76 89.50 Brown B
91 161.75 89.30 Blue B
92 168.40 80.54 Green A
93 164.69 90.93 Blue A
94 164.61 81.30 Blue A
95 168.25 69.42 Grey A
96 172.41 68.56 Blue B
97 170.50 82.31 Blue A
98 177.90 66.75 Green A
99 170.18 91.41 Green B
100 176.35 88.30 Grey B
101 163.61 79.84 Grey A
102 170.37 83.49 Blue B
103 162.80 83.93 Grey A
104 165.08 100.79 Green B
105 168.06 77.16 Grey A
106 178.30 87.79 Blue A
107 158.56 74.17 Brown B
108 176.90 90.69 Grey A
109 165.05 83.44 Brown A
110 176.26 88.39 Brown B
111 164.35 94.24 Brown B
112 162.52 89.67 Brown B
113 163.37 82.60 Brown B
114 168.13 85.49 Brown A
115 167.09 82.06 Blue A
116 166.98 75.99 Green A
117 170.33 80.69 Brown B
118 168.70 79.16 Grey B
119 171.32 66.15 Blue A
120 178.29 79.79 Blue A
121 176.63 93.71 Grey B
122 170.27 85.53 Grey A
123 168.90 68.24 Brown A
124 174.65 54.62 Green A
125 173.43 71.15 Brown B
126 163.25 84.85 Blue B
127 173.17 74.42 Blue B
128 159.67 69.37 Brown B
129 170.33 88.54 Brown A
130 176.08 70.47 Blue B
131 165.73 87.17 Brown B
132 175.71 64.11 Brown A
133 158.95 90.90 Brown A
134 172.04 94.32 Brown A
135 176.93 82.80 Grey A
136 166.87 94.96 Brown A
137 168.25 83.90 Green A
138 166.54 76.47 Grey B
139 178.04 75.91 Green A
140 171.63 85.71 Brown A
141 174.82 87.38 Brown B
142 170.00 69.44 Blue A
143 165.06 91.37 Blue B
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400 168.62 56.65 Brown A
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402 174.01 91.69 Grey A
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602 169.09 66.55 Blue A
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786 163.61 80.50 Blue A
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823 172.03 94.45 Blue A
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829 163.88 87.71 Grey A
830 178.25 65.16 Blue A
831 169.10 84.58 Brown A
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834 163.67 78.18 Grey A
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836 167.80 85.85 Blue B
837 166.86 73.36 Brown A
838 179.33 75.85 Green B
839 164.99 81.11 Blue A
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843 167.13 69.47 Blue B
844 165.07 90.08 Grey B
845 169.01 58.07 Blue B
846 171.05 81.41 Grey A
847 164.38 84.70 Grey B
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851 179.47 90.55 Green B
852 180.05 57.27 Brown B
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855 166.17 83.05 Brown A
856 172.60 84.16 Blue A
857 175.95 73.06 Green B
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859 164.43 52.49 Green B
860 180.30 77.35 Brown A
861 168.77 67.67 Grey B
862 154.85 83.16 Brown B
863 167.43 99.05 Green B
864 153.55 71.55 Grey A
865 158.52 85.64 Grey A
866 166.01 69.37 Brown B
867 164.43 76.99 Grey B
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871 171.12 81.33 Grey B
872 163.77 76.05 Grey A
873 175.04 77.48 Blue A
874 176.59 102.74 Green B
875 169.23 67.34 Green A
876 165.06 84.15 Brown A
877 169.10 72.25 Green A
878 173.07 88.73 Brown A
879 171.58 72.35 Brown B
880 170.34 69.75 Brown A
881 157.07 91.28 Green B
882 174.68 86.03 Grey B
883 169.55 105.16 Green A
884 172.26 69.65 Brown A
885 166.95 102.74 Blue B
886 173.16 67.80 Grey B
887 163.87 93.58 Green A
888 164.05 53.66 Blue B
889 175.62 83.99 Blue A
890 175.04 90.40 Blue A
891 175.00 69.03 Blue B
892 163.12 83.92 Grey A
893 160.41 81.10 Grey A
894 164.59 76.25 Grey A
895 170.98 81.28 Brown B
896 176.30 64.79 Brown B
897 174.58 77.13 Green A
898 164.00 85.53 Brown A
899 158.81 73.95 Brown B
900 176.49 77.54 Blue A
901 169.66 68.72 Grey B
902 160.50 73.35 Brown A
903 164.66 94.38 Green B
904 170.40 90.15 Blue A
905 163.80 96.31 Grey B
906 168.98 78.93 Grey A
907 179.11 84.44 Green A
908 169.20 69.87 Green A
909 170.75 84.63 Grey A
910 171.31 84.22 Green A
911 175.38 81.53 Grey B
912 173.87 73.93 Grey A
913 167.49 105.02 Blue B
914 170.70 70.27 Green B
915 167.82 77.02 Brown B
916 172.47 90.07 Grey B
917 166.75 89.11 Brown A
918 180.36 66.44 Green B
919 168.75 92.93 Grey B
920 170.89 101.59 Brown A
921 175.85 75.51 Grey B
922 180.04 84.26 Blue B
923 171.04 78.29 Blue B
924 153.51 72.37 Brown A
925 177.84 66.64 Brown A
926 173.67 85.03 Blue A
927 169.33 89.97 Grey B
928 157.66 75.72 Green B
929 168.23 88.40 Grey B
930 163.20 112.01 Brown A
931 166.49 84.73 Blue B
932 164.94 55.95 Brown B
933 184.23 77.38 Grey A
934 163.89 57.70 Blue A
935 177.42 88.59 Green A
936 165.40 67.20 Brown A
937 173.63 69.10 Brown B
938 172.05 105.04 Grey B
939 169.04 89.07 Grey A
940 172.33 74.63 Green B
941 163.11 81.14 Green B
942 167.25 80.97 Brown A
943 171.85 84.04 Brown B
944 171.15 94.66 Blue B
945 172.66 66.67 Grey B
946 161.80 75.28 Blue A
947 166.41 89.42 Green A
948 170.99 80.73 Blue A
949 173.01 87.67 Grey B
950 171.35 66.75 Grey A
951 160.63 91.08 Blue B
952 174.08 66.82 Blue A
953 172.22 89.78 Grey A
954 172.57 90.79 Blue A
955 172.31 75.52 Grey A
956 178.45 66.69 Blue B
957 168.67 84.34 Blue A
958 162.35 68.54 Blue B
959 168.17 58.81 Green A
960 171.66 78.64 Green B
961 172.03 69.68 Blue A
962 176.08 85.25 Brown B
963 175.58 64.86 Green B
964 173.74 77.21 Grey B
965 162.38 56.99 Green A
966 177.40 67.66 Green B
967 173.26 70.92 Blue A
968 169.47 86.97 Grey A
969 164.55 101.67 Green B
970 171.09 95.36 Blue A
971 168.34 79.49 Grey B
972 173.36 75.20 Green B
973 173.69 72.27 Grey A
974 175.67 79.14 Grey B
975 163.98 91.93 Green B
976 170.40 89.87 Green B
977 161.75 71.14 Brown B
978 174.45 97.85 Brown B
979 164.01 87.02 Brown A
980 168.01 72.12 Green A
981 166.01 70.70 Green A
982 164.53 70.62 Grey A
983 167.30 72.98 Brown B
984 172.11 99.68 Green B
985 172.45 84.12 Grey B
986 169.16 78.10 Grey B
987 168.53 87.21 Green A
988 176.67 81.50 Green B
989 169.90 95.50 Brown A
990 171.53 77.40 Blue B
991 171.34 72.32 Green B
992 177.02 76.03 Grey B
993 166.97 70.67 Blue B
994 170.55 95.25 Blue B
995 163.57 69.84 Brown A
996 179.04 77.39 Blue B
997 168.27 74.75 Green B
998 178.57 96.92 Green A
999 162.76 72.21 Grey A
1000 164.82 80.71 Brown A

Check how similar they are

Increasingly similar

In fact, if we did this many, many, many times, these groups would be, on average, increasingly identical!

Why?

  • Central limit theorem

  • Law of large numbers

Do international monitors deter election-day fraud?

Yes!

  • The international community monitored the 2003 Armenian Presidential elections
  • Monitors were assigned randomly to the polling stations
  • Hyde found a big average difference between the vote share received by the cheating party at monitored stations compared to non-monitored stations.